Notation

The following notation is helpful, when we talk about the
standard deviation and the standard error.

Population parameter

Sample statistic

N: Number of observations in the population

n: Number of observations in the sample

Ni: Number of observations in population
i

ni: Number of observations in sample
i

P: Proportion of successes in population

p: Proportion of successes in sample

Pi: Proportion of successes
in population i

pi: Proportion of successes
in sample i

μ: Population mean

x: Sample estimate of
population mean

μi: Mean of population i

xi: Sample estimate of
μi

σ: Population standard deviation

s: Sample estimate of σ

σp: Standard deviation of p

SEp: Standard error of p

σx:
Standard deviation of x

SEx:
Standard error of x

Standard Deviation of Sample Estimates

Statisticians use sample statistics to estimate population
parameters.
Naturally, the value of a statistic may vary from one sample
to the next.

The variability of a statistic is measured by its standard deviation.
The table below shows formulas for computing the standard deviation of
statistics from
simple random samples. These formulas are valid when the
population size is much larger (at least 10 times larger) than
the sample size.

Statistic

Standard Deviation

Sample mean, x

σx =
σ / sqrt( n )

Sample proportion, p

σp =
sqrt [ P(1 - P) / n ]

Difference between means,
x1 -
x2

σx1-x2 =
sqrt [ σ21 / n1 +
σ22 / n2 ]

Difference between proportions,
p1 -
p2

σp1-p2 =
sqrt [ P1(1-P1) / n1 +
P2(1-P2) / n2 ]

Note: In order to compute the standard deviation of a sample
statistic, you must know the value of one or more
population parameters.

Standard Error of Sample Estimates

Sadly, the values of population parameters are often unknown, making
it impossible to compute the standard deviation of a statistic.
When this occurs, use the standard error.

The standard error is computed from known sample statistics, and it
provides an unbiased estimate of the standard deviation of the statistic.
The table below shows how to compute the standard error for
simple random samples, assuming the population size is
at least 10 times larger than the sample size.

Statistic

Standard Error

Sample mean, x

SEx =
s / sqrt( n )

Sample proportion, p

SEp =
sqrt [ p(1 - p) / n ]

Difference between means,
x1 -
x2

SEx1-x2 =
sqrt [ s21 / n1 +
s22 / n2 ]

Difference between proportions,
p1 -
p2

SEp1-p2 =
sqrt [ p1(1-p1) / n1 +
p2(1-p2) / n2 ]

The equations for the standard error are identical to the equations
for the standard deviation, except for one thing - the standard
error equations use statistics where the standard deviation
equations use parameters. Specifically, the standard error
equations use p in place of P, and
s in place of σ.

Test Your Understanding

Problem 1

Which of the following statements is true.

I. The standard error is computed solely from sample attributes.
II. The standard deviation is computed solely from sample attributes.
III. The standard error is a measure of central tendency.

(A) I only
(B) II only
(C) III only
(D) All of the above.
(E) None of the above.

Solution

The correct answer is (A). The standard error can be computed
from a knowledge of sample attributes - sample size and sample
statistics. The standard deviation cannot be computed
solely from sample attributes; it requires a knowledge of one or
more population parameters. The standard error is a measure of
variability, not a measure of central tendency.